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Let R be a commutative ring with identity. 43] that (S = Pic(R), ⊗) is a symmetric monoidal category. Since π0 (S) is a group, S and S −1 S are homotopy equivalent (see [167]). Hence we get K0 Pic(R) = Pic(R), K1 (Pic(R)) = U (R) (units of R), and Kn (Pic(R)) = 0 for all n 2. 20 A. 3. 1. D EFINITION . A category with cofibrations is a category C with zero object together with a subcategory co(C) whose morphisms are called cofibrations written A B and satisfying the axioms (C1) Every isomorphism in C is a cofibration.

Matrices of the form 1 + N for some nilpotent matrix N . 14 A. 3. T HEOREM [151]. 1 = K (R[Δq ]) ⇒ K k−v (R). (i) For p 1, q 0, there is a spectral sequence Epq p p+q (ii) If R is regular, then the spectral sequence in (i) above degenerates and Kn (R) = Knk−v (R) for all n 1. 4. D EFINITION . ) is said to be homotopy invariant if for any ring R, the natural map R → R[t] induces an isomorphism F (R) ≈ F (R[t]). Note that if F is homotopy invariant, then the simplicial object F (R[Δ∗ ]) is constant.

If X = Spec(A), then we have an equivalence of categories M(X) ≈ M(A) and Gn (X) ≈ Gn (A). (v) Let R be a commutative ring with identity, Λ an R-algebra that is finitely generated as an R-module, PR (Λ) the category of left Λ-lattices. Then PR (Λ) is an exact category and we write Gn (R, Λ) for Kn (PR (Λ)). If Λ = RG, G finite group, write Gn (R, G) for Gn (R, RG). If R is regular, then Gn (R, Λ) ≈ Gn (Λ), see [67]. 7(ii). Then, the category [S, C] of functors from S to an exact category C is also an exact category.